Given the functions:

f(x)=2x+2 g(x)=3/x^2+2 h(x)=sqrt(x-2)

Find the following
1) (f∘g)(-5)
2) (h∘g)(3)
3) (g∘h)(-2)
4) (g∘g)(-2)
5) (f∘h)(12)

Find the following and state the domain:
6) h∘g
7) f∘g
8) g∘f
9) g∘g
10)g∘h

I can't edit but f(x) is suppose to be f(x)=3x+2 I put 2x by mistake!

I'll do one. Do the others in like wise. If you get stuck, come on back with what you did.

(g∘h) = 3/(h^2+2) = 3/(x-2+2) = 3/x
(g∘h)(-2) = 3/-2 = -3/2

The domain of h is x>=2
The domain of g is all reals, since x^2+2 is never zero.
So, the domain of (g∘h) is all reals x >= 2 because you can't have (g∘h)(x) unless h is defined.

To find the composition of functions, you substitute the inner function into the outer function. Let's go through each question one by one.

1) To find (f∘g)(-5), substitute -5 into g(x) first, then substitute the result into f(x).
Step 1: g(-5) = 3/(-5)^2 + 2 = 3/25 + 2 = 53/25
Step 2: f(53/25) = 2 * (53/25) + 2 = 106/25 + 2 = 106/25 + 50/25 = 156/25

So, (f∘g)(-5) = 156/25

2) To find (h∘g)(3), substitute 3 into g(x) first, then substitute the result into h(x).
Step 1: g(3) = 3/(3)^2 + 2 = 3/9 + 2 = 1/3 + 2 = 7/3
Step 2: h(7/3) = sqrt(7/3 - 2) = sqrt(7/3 - 6/3) = sqrt(-5/3) (Domain error)

In step 2, we get a domain error because the square root of a negative number is not defined in the domain of real numbers. Therefore, (h∘g)(3) is undefined.

3) To find (g∘h)(-2), substitute -2 into h(x) first, then substitute the result into g(x).
Step 1: h(-2) = sqrt(-2 - 2) = sqrt(-4)

In step 1, we encounter a domain error because the square root of a negative number is not defined in the domain of real numbers. Therefore, (g∘h)(-2) is undefined.

4) To find (g∘g)(-2), substitute -2 into g(x) twice.
Step 1: g(-2) = 3/(-2)^2 + 2 = 3/4 + 2 = 11/4
Step 2: g(11/4) = 3/(11/4)^2 + 2 = 3/(121/16) + 2 = 48/121 + 2 = 48/121 + 242/121 = 290/121

So, (g∘g)(-2) = 290/121

5) To find (f∘h)(12), substitute 12 into h(x) first, then substitute the result into f(x).
Step 1: h(12) = sqrt(12 - 2) = sqrt(10)
Step 2: f(sqrt(10)) = 2 * sqrt(10) + 2

So, (f∘h)(12) = 2 * sqrt(10) + 2

Now let's move on to the next set of questions:

6) To find h∘g, substitute g(x) into h(x) directly.
h∘g(x) = sqrt(g(x) - 2) = sqrt((3/x^2 + 2) - 2)

The domain of h(x) is x > 2, and the domain of g(x) is all real numbers except x = 0. Therefore, the domain of h∘g is the intersection of these two domains, which is x > 2, excluding x = 0.

7) To find f∘g, substitute g(x) into f(x) directly.
f∘g(x) = 2 * g(x) + 2 = 2 * (3/x^2 + 2) + 2 = 6/x^2 + 4 + 2 = 6/x^2 + 6

The domain of g(x) is all real numbers except x = 0. Therefore, the domain of f∘g is the same as the domain of g(x), which is all real numbers except x = 0.

8) To find g∘f, substitute f(x) into g(x) directly.
g∘f(x) = 3/(f(x))^2 + 2 = 3/(2x+2)^2 + 2 = 3/(4x^2 + 8x + 4) + 2

The domain of f(x) is all real numbers. Therefore, the domain of g∘f is also all real numbers.

9) To find g∘g, substitute g(x) into g(x) directly.
g∘g(x) = 3/(g(x))^2 + 2 = 3/(3/x^2 + 2)^2 + 2

The domain of g(x) is all real numbers except x = 0. Therefore, the domain of g∘g is the same as the domain of g(x), which is all real numbers except x = 0.

10) To find g∘h, substitute h(x) into g(x) directly.
g∘h(x) = 3/(h(x))^2 + 2 = 3/(sqrt(x-2))^2 + 2 = 3/(x-2) +2

The domain of h(x) is x > 2. Therefore, the domain of g∘h is x > 2.